The theory of determinants in the historical order of development, by Sir Thomas Muir.

ORTHOGONANTS (SYLVESTER, 1878)

303'

SYLVESTER, J. J. (1878).
[Preuve instantanee d'apres la methode de Fourier de la realite des
racines de l'equation seculaire. Crelle's Journ., lxxxviii.
pp. 4-5; or Collected Math. Papers, iii. pp. 451-452.]
The proof in question is only a slight variant of Salmon's of 1859,.
which again, as we have seen, is in essence not really different from
Cauchy's of 1829.
Following upon it is a short paper (pp. 6-9) intended to show that
the property possessed by the determinant in question is shared by
one that, though still axisymmetric, is much more general.
Sylvester's first step in this direction had been taken twenty-five
years before (Hist., ii. p. 314), and Cauchy's two years later.
SOURANDER, E. (1879).
[Sur l'equation dont dependent les inegalites seculaires des planetes.
Journ. (de Liouville) de Math., (3) v. pp. 195-208; or Acta
Soc. Sci. Penn. (Helsingfors), xi. pp. 257-271.]
Sourander, like Bauer, takes the product of the squared differences
of the roots of the equation
a-x h g
h b-x f = 0, or, say, x3-Px2+Qx-R = 0,,
g f c-x
in the form
2(Q2-3RP) -PQ+9R
-PQ+9R 2(P2-3Q)
but, unlike Bauer and improving on him, he effects the transforma —
tion of it into 12 times the square of a 2-by-6 array, and therefore
into 12 times the sum of the squares of fifteen determinants of the
second order. The essential part of his procedure consists in.
introducing
b-c c-a a-b
u, v, w }6 ' /6' 16
U V W to stand for B-C C-A A -
/16 S' 6 ' 6